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HMMT 二月 2026 · ALGNT 赛 · 第 1 题

HMMT February 2026 — ALGNT Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A line intersects the graph of y = x + at three distinct points. Given that the x -coordinates of two x of the points are 6 and 7 , respectively, compute the x -coordinate of the third point.
解析
  1. A line intersects the graph of y = x + at three distinct points. Given that the x -coordinates of two x of the points are 6 and 7 , respectively, compute the x -coordinate of the third point. Proposed by: Pitchayut Saengrungkongka 1 Answer: − 21 Solution: Let the equation of the line be y = ax + b . Then, the x -coordinates of the intersection points satisfy 2 2 ax + b = x + , x 3 2 or x − ax − bx + 2 = 0 . We know that two of the solutions to this equation are x = 6 and x = 7 . Furthermore, the product of all three solutions is − 2 by Vieta’s formula. Hence, the third solution is − 2 1 = − . 6 · 7 21